CVEN 4402 Workshop Week 7 Solution.pdf

CVEN 4402 Workshop Week 7 Solution.pdf - CVEN 4402 Network...

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CVEN 4402 Network Analysis Workshop Week 7 User Equilibrium Solution Methods: The Franke-Wolfe Method
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Question 1 Frank Wolfe Hints and Help The outline of the FW algorithm is as follows: 1. Choose an initial solution, x 0 2. For iteration i , evaluate the gradient of the objective function at x i-1 , which will be 𝛻? ? ?−1 3. Find the new vector ? that minimizes the product 𝛻? ? ? subject to the original constraints. 4. Find the value of 𝝀 that minimizes the expression 𝒇(𝝀𝒙 + ? − 𝝀 𝒙 𝒊−? ) using bisection method or golden section method. 5. Repeat 2-4 until convergence. Consider the following problem: ?𝐢? 𝒇 𝒙 = (𝒙 − ?) ? s.t. −?𝟓 ≤ 𝒙 ≤ ?? Solve this problem using MSA and the Frank Wolfe algorithm .
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Frank Wolfe Iteration 1 Step 1: Choose starting place x 0 = 0 Step 2: Evaluate gradient at x 0 Step 3: Minimize product Step 4: Find 𝜆 that minimizes expression below, to determine x 1 𝛻? ? = 2 ? − 2 𝛻? ? 0 = 2 0 − 2 = −4 min 𝛻? ? ? = −4 ? s.t. − 15 ≤ ? ≤ 10 ? = 10 ?(𝜆? + 1 − 𝜆 ? i−1 ) ? ? = 𝜆? + 1 − 𝜆 ? i−1
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Step 4: Optimal! Frank Wolfe Iteration 1 ? ?−1 = 0 and ? = 10 ?(𝜆? + 1 − 𝜆 ? i−1 ) = (𝜆? + 1 − 𝜆 ? 0 ) − 2 2 = 10λ − 2 2 𝜆 = 0.2 → ? 1 = 𝜆? + 1 − 𝜆 ? 0 = 0.2 × 10 + 1 − 0.2 × 0 = 2
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UE Mathematical Programming Formulation and Optimality Conditions ?𝑖?𝑖?𝑖?? ? 𝑎 (?)?? ? 𝑎 0 ∀𝑎 s.t. ? 𝑘 ?? 𝑘 = ? ?? ∀? , ? ? 𝑘 ?? 0 ∀𝑘 , ? , ? ? 𝑎 = ? 𝑘 ?? 𝛿 𝑎𝑘 ?? 𝑘 ? ? ∀𝑎 This is the objective function “subject to” These are the constraints **This entire formulation (objective function + constraints) is a “program” (can be called other things as well!) A means “for each” t a is the cost function on link a f is the flow on path k for OD pair rs q is the demand for OD pair rs x is the flow on link a
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